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My question concerns a household's utility function U(c,l). We're told that the partial derivatives dU/dc and dU/dl > 0, and the second order partial derivatives are negative. Now, can I assume anything about the sign of the partial derivatives Ucl (or Ulc)?

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No.

In general, based on the information you have, there are no results you can infer for the partial derivatives $U_{cl}$ / $U_{lc}$. In this context they could be anything.

As to what you are allowed to assume is a different story. You can assume anything in general, of course. Nevertheless, there is no way to check if your assumptions would be correct in this case.

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For a "well-behaved" bivariate utility function as regards utiliy maximiaztion, we need the Hessian matrix of partial derivatives to be negative definite, which translates into

$$U_{cc}<0, \;\;\;U_{ll}<0,\;\;\; U_{cc}U_{ll} - \left[U_{cl}\right]^2 >0$$

$$\implies |U_{cl} | < \sqrt{U_{cc}U_{ll}}$$

The first two conditions you have by assumption, the third condition poses a constraint on the relative magnitude in absolute value of the cross-partial, although not on its sign: it has to be smaller in absolute terms than the geometric average of the own-second derivatives, at least at the optimal point.

I notice that in many research papers it is assumed that the cross-partial is zero (utility additive in consumption and leisure).

Nevertheless, this is a good opportunity to apply some economic/behavioral thinking: what would it mean, as regards the relation between consumption and leisure to have $U_{cl} <0$? What would it mean to have $U_{cl} >0$ ?

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